Markov set-valued functions on compact metric spaces (Q6591284)

\textit{S. Holte} [Topology Appl. 123, 421--427 (2002; Zbl 1010.37020)] introduced the notion of Markov functions on closed intervals \([a, b]\) as follows: The Markov partition of a closed interval \(I = [0, 1]\) with respect to a continuous function \(f : I\rightarrow I\) is usually given by finitely many points \(0 = x_0 < x_1 < x_2 < \cdots < x_{n-1} < x_n = 1\) in \(I\) such that all the restrictions \(f |_{[x_{i-1},x_i]}\) of \(f\) to \([x_{i-1}, x_i]\) are homeomorphisms from \([x_{i-1}, x_i]\) onto some interval \([x_k, x_l]\). If a continuous function \(f\) has a Markov partition \(A\), then we say that \(f\) is a Markov function with respect to \(A\). Also, she proved the following theorem:\N\NTheorem. Let \(I\) and \(J\) be closed intervals, let \(f\) be a Markov function with respect to \(A \subset I\) and let \(g\) be a Markov function with respect to \(B \subset J\). If \(f\) and \(g\) follow the same pattern then the inverse limits \(\displaystyle{\lim_{\longleftarrow} (I,f_n)}\) and \(\displaystyle{\lim_{\longleftarrow} (J,g_n)}\) are homeomorphic. \N\NIn the present paper, the authors generalize the notion of Markov functions on closed intervals \([a, b]\) to Markov set-valued functions on compact metric spaces. They also generalize Holte's result when two such Markov set-valued functions follow the same pattern and show that if the Markov set-valued functions \(F : X \multimap X\) and \(G : Y\multimap Y\) follow the same pattern, then the inverse limits \(\displaystyle{\lim_{\multimap} (X,F)}\) and \(\displaystyle{\lim_{\multimap} (Y,G)}\) are homeomorphic.